solution:
s( t ) = ( 2 )sin( πt ) + ( 2 )cos(πt )
v( t ) = s'( t ) = ( 2π )cos( πt) - ( 2π )sin( πt )
vavg = 1 / ( b - a ) Integral a to b [ v( t ) ] dt
( a )
vavg
= 1 / ( 2 - 1 ) Integral 1 to 2 [ ( 2π)cos( πt ) - ( 2π )sin( πt ) ] dt
= [ ( 2)sin( πt ) + ( 2 )cos( πt ) ] 1to 2
= [ ( 2)sin( 2π ) + ( 2 )cos( 2π ) ] - [ ( 2 )sin( π ) + ( 2 )cos( π) ]
= 4 cm / s
( b )
vavg
= 1 / ( 1.1 - 1 ) Integral 1 to 1.1 [ ( 2π)cos( πt ) - ( 2π )sin( πt ) ] dt
= 10 [ ( 2 )sin( πt ) + ( 2 )cos( πt ) ] 1 to 1.1
= 10 [ [ ( 2 )sin( 1.1π ) + ( 2)cos( 1.1π ) ] - [ ( 2 )sin( π) + ( 2 )cos( π ) ] ]
-5.20 cm /s
( c )
vavg
= 1 / ( 1.01 - 1 ) Integral 1 to 1.01 [ ( 2π)cos( πt ) - ( 2π )sin( πt ) ] dt
= 100 [ ( 2 )sin( πt ) + ( 2 )cos( πt ) ] 1 to 1.01
= 100 [ [ ( 2 )sin( 1.01π ) + (2 )cos( 1.01π ) ] - [ ( 2 )sin( π) + ( 2 )cos( π ) ] ]
-6.18 cm /s
( d )
vavg
= 1 / (1.001 - 1 ) Integral 1 to 1.001 [ ( 2π )cos( πt ) - ( 2π )sin(πt ) ] dt
= 1000 [ ( 2 )sin( πt ) + ( 2 )cos( πt ) ] 1 to 1.001
= 1000 [ [ ( 2 )sin( 1.001π ) + (2 )cos( 1.001π ) ] - [ ( 2 )sin(π ) + ( 2 )cos( π ) ] ]
-6.27 cm /s
Did you you get the answer to this one?
Answer:
8/19 or 42.1
Step-by-step explanation:
<span>The fact that Helen’s indifference curves touch the axes should immediately make you want to check for a corner point solution. To see the corner point optimum algebraically, notice if there was an interior solution, the tangency condition implies (S + 10)/(C +10) = 3, or S = 3C + 20. Combining this with the budget constraint, 9C + 3S = 30, we find that the optimal number of CDs would be given by 3018â’=Cwhich implies a negative number of CDs. Since it’s impossible to purchase a negative amount of something, our assumption that there was an interior solution must be false. Instead, the optimum will consist of C = 0 and Helen spending all her income on sandwiches: S = 10. Graphically, the corner optimum is reflected in the fact that the slope of the budget line is steeper than that of the indifference curve, even when C = 0. Specifically, note that at (C, S) = (0, 10) we have P C / P S = 3 > MRS C,S = 2. Thus, even at the corner point, the marginal utility per dollar spent on CDs is lower than on sandwiches. However, since she is already at a corner point with C = 0, she cannot give up any more CDs. Therefore the best Helen can do is to spend all her income on sandwiches: ( C , S ) = (0, 10). [Note: At the other corner with S = 0 and C = 3.3, P C / P S = 3 > MRS C,S = 0.75. Thus, Helen would prefer to buy more sandwiches and less CDs, which is of course entirely feasible at this corner point. Thus the S = 0 corner cannot be an optimum]</span>
